A further remark on the density estimate for degenerate Allen-Cahn equations: $Δ_{p}$-type equations for $1<p<\frac{n}{n-1}$ with rough coefficients
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| Format: | Preprint |
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2025
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| _version_ | 1866909859822174208 |
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| author | Zhang, Chilin |
| author_facet | Zhang, Chilin |
| contents | In this short remark on a previous paper \cite{SZ25}, we continue the study of Allen-Cahn equations associated with Ginzburg-Landau energies \begin{equation*}
J(v,Ω)=\int_Ω\Big\{F(\nabla v,v,x)+W(v,x)\Big\}dx, \end{equation*} involving a Dirichlet energy $F(\vecξ,τ,x)\sim|\vecξ|^{p}$ and a degenerate double-well potential $W(τ,x)\sim(1-τ^{2})^{m}$. In contrast to \cite{SZ25}, we remove all regularity assumptions on the Ginzburg-Landau energy. Then, with further assumptions that $1<p<\frac{n}{n-1}$ and that $W(τ,x)$ is monotone in $τ$ on both sides of $0$, we establish a density estimate for the level sets of nontrivial minimizers $|u|\leq1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19820 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A further remark on the density estimate for degenerate Allen-Cahn equations: $Δ_{p}$-type equations for $1<p<\frac{n}{n-1}$ with rough coefficients Zhang, Chilin Analysis of PDEs Mathematical Physics In this short remark on a previous paper \cite{SZ25}, we continue the study of Allen-Cahn equations associated with Ginzburg-Landau energies \begin{equation*} J(v,Ω)=\int_Ω\Big\{F(\nabla v,v,x)+W(v,x)\Big\}dx, \end{equation*} involving a Dirichlet energy $F(\vecξ,τ,x)\sim|\vecξ|^{p}$ and a degenerate double-well potential $W(τ,x)\sim(1-τ^{2})^{m}$. In contrast to \cite{SZ25}, we remove all regularity assumptions on the Ginzburg-Landau energy. Then, with further assumptions that $1<p<\frac{n}{n-1}$ and that $W(τ,x)$ is monotone in $τ$ on both sides of $0$, we establish a density estimate for the level sets of nontrivial minimizers $|u|\leq1$. |
| title | A further remark on the density estimate for degenerate Allen-Cahn equations: $Δ_{p}$-type equations for $1<p<\frac{n}{n-1}$ with rough coefficients |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2507.19820 |